...why this system exists
If the music is essentially diatonic - like the past 1,000 years of western musical tradition - then the seven letters
ABCDEFG (the gamut) represent the diatonic tones. The sharps and flats of key signatures then represent transpositions of the diatonic gamut. Accidentals represent what can be through of as momentary transpositions (secondary chords) or modal/chromatic alterations of the essential diatonic gamut, for example, the raised/lowered seventh scale degree in minor, or the "double" leading tone chromaticism of augmented sixth chords.
I find it frustrating that all systems I’ve seen use the accidental system
That's because so much music is essentially diatonic expanded with some elements of chromaticism.
You are in the key of F# major and need key change to a sharper key
Now you're getting into a specific situation.
If you are in a key signature with lots of sharps and flats, you can get accidentals of double sharps and double flats, or other spellings which can be hard to read. Some of that is connected to music history and convention. In the classical era key signatures tended to use less sharps/flats, and modulations were to "close" keys. That stuff isn't really hard to read. Later periods pushed into more distant keys and more chromaticism. Certain styles conventionally use simple keys. I sight read from hymnals, and even though some are from more recent times, they tend to use simple key signatures. You are not acknowledging that the standard notation system is easy to read for a huge amount of existing music.
My reading skills are not that great, but a big part of the issue is reading relative harmonic changes, and getting a sense for what the accidentals provide as harmonic queues to typical patterns. For example, if I see a sharp, I usually recognize it as a temporary leading tone and my hands move for some kind of
V I movement which you technically might call tonicization or think of as a sort of localized transposition. So, if the music were in
C major and I see a
G♯, I think
A minor and sort of shift my mental and hand orientation to
A minor temporarily. Similarly, if I see a flat, I will probably assume the flat is on the seventh scale degree and the move is to the subdominant which would be like playing in
C major, hitting a
B♭, and moving to an
F chord. The point being that
A minor, and
F major are all diatonic, and the accidentals are both queues to (sort of) transpose and they are the points where the various keys "overlap".
I need to name the letter is the above example just to be clear, but in actual reading I'm not paying attention to each and every letter. Realtive changes and staff line distances are much more important. You can't avoid reading letters entirely, because you need some point of reference. It becomes more like this, when I get oriented to the
A, let's say the second space on the treble clef staff, then when I see a
♯ on the line below, I don't really care that it's a
♯ is really telling me the note below
A is a half step below. Similarly, if the key were
F♯ and I'm playing a note on the second line of the treble staff and the next note is on the space below with a double sharp
x, I really try to not think "F double sharp" but recognize the note on the space below with
x as "a half step below" the
G♯ I'm playing.
Have you ever seen when someone has written the letter names on keyboard keys, or maybe even on a fretboard? That's a bad idea if you really want to handle chromatic harmony. The key someone might naively mark as
G is really a
G, or a
Fx or an
None of that will make much sense or be do-able until one has practices all major and minor scales, along with various harmonic patterns (all the cadences, circle of fifth sequences, rule of the octave, etc.), in three inversions/positions, in all the major and minor keys. You have to practice that stuff until you are equally comfortable playing basic patterns in any key. You should be able to, for example, hit and resolve a French augmented sixth chord to the dominant, in all inversion, in any key, without hesitation.
It may seem contradictory to some of what I've written above, but when practicing patterns in all keys you do what to think of the pitch letters, because certain spelling make more sense in particular harmonic contexts. For example, go up three half steps from middle
C. As a major key tonic, that will normally be
E♭. If I play a major triad on that tone, I tend to always think of it as
E♭ major, but this is wrong. If the key is
E♭ major, then sure, the chord is
E♭ major. But, if the key were
G♯ minor, that same triad should be properly called
D♯ major. So, when practicing in
G♯ minor, I will be very careful to mentally recognize
D# and not
E♭. Similar thing for the leading tone in
G♯ minor, it's
Fx and not
...artificial problems (ex. You are in the key of F# major and need key change to a sharper key)
I'm not sure I see what the problem is.
F♯ major is six sharps. "Sharper" I suppose is one more sharp in the key signature,
C♯ major, seven sharps. You could enharmonically change that to
D♭ major, 5 flats to keep things simpler. Theoretically you can keep adding sharps to the key signature. From
C♯ major you could go to
G♯ major, all seven letters sharped, but the
F would be double sharp. That's theoretical, but a practical nightmare, so you enharmonically respell
G♯ major to
A♭ major. For all practical purposes the worst you have to deal with is key signatures of either seven sharps or seven flats. The dilemma for reading is getting used to reading double sharps and flats. I think the path to that skill is not reading discrete letter spellings, but relative changes where most of those accidentals are raised secondary leading tones or leading tones lowered to become subdominant degrees.
Is there any note naming system out there that uses the normal note names for non-accidental notes (A,B,C,D,E,F,G) but has one syllable unique names for the accidental notes?
There are chromatic solfege systems that do exactly that, give unique syllables to the 12 chromatic tones.
But those are not notation systems.
I could be wrong, but I suspect the real "problem" for you may be reading staff notation rather than the pitch naming system. A chromatic 12 syllable system won't resolve that problem.